Arithmetic. (Article. February 1896.)
VIII - MULTIPLICATION OF FRACTIONS - CONTINUED
Care should be exercised in the choice of fractions. If the pupil has been thoroughly taught cancellation, possibly he will lost a little in his attempt to learn the explanation, provided he is able to do more or less canceling. In some instances we have thought it advisable to treat the multiplication of a fraction by a fraction as a case in cancellation. We are inclined to think this might be done successfully, and this such an explanation as we have already offered would be unnecessary. After the pupils have worked out step by step, the explanation of a dozen or more problems, a generalization may be made. The pupil is led to observe that in multiplying 4-5 by 3-7, he has, in reality, multiplied the numerators of the two fractions for a new numerator, and the denominators of the two fractions for a new denominator. He thus makes for himself the rule that is commonly stated in our books for multiplying a fraction by a fraction.
Again we take the liberty to emphasize the importance of compelling pupils to work a very large number of problems involving the preceding cases, having the pupils bear in mind that the last case is of no special value except as it becomes part of the preceding cases. We are aware that this work is not especially attractive to the teacher. We do ask, however, for a thoughtful and thorough test of the plan herein set forth.
DIVISION OF FRACTIONS
The same simplicity cannot be insisted upon in division of fractions. For his first case the pupil may be asked to divide 450 ¾ by seven, or any other number that he can use as a divisor, and work, as he commonly terms it, by short division. The teacher should not underestimate the importance of doing this work. The truth of the matter is, the majority of problems in division of fractions involve just this case. Already the reader has thought out what the second case should be, provided a rigid attempt is made to be logical.
The pupil would next be asked to divide 420 by 7 ⅔. This he can no easily do, nor can he be taught to do it without learning something that is radically new. Even the effort that some teachers make to introduce decimal fractions would not in this problem be of any value. We have found the easiest plan to be the following: The pupil is led to change both dividend and divisor to thirds. Then his problem is a problem in integers. He has applied a principle of simple division. He has multiplied both dividend and divisor by the same number, and knows that his quotient is not thereby changed.
We would be very glad to have every reader write us in relation to this particular case. We would be very glad to ascertain what business men do, as a rule, in a problem like this. If this plan is adopted and followed generally there are no more difficulties to be encountered in division of fractions, because the dividing of an integer and a fraction by an integer and a fraction would fall under this method. There are readers, however, who will present the question that is so often asked at Institutes: "In division of fractions do you teach pupils to invert the terms of the divisor?" This depends upon how you wish the pupil to divide an integer and a fraction by an integer and a fraction, or how you wish the pupil to divide a fraction by a fraction. The thoughtful reader will at once observe that he can change both dividend and divisor to the same denomination and proceed as in integers; or if, for any reason, the teacher wishes to adopt the method suggested in many of our books, he may proceed as follows: Divide 4-5 by 3-7. According to preceding principle, to divide a fraction by an integer divide the numerator or multiply the denominator. In this case we will multiply the denominator. Doing this we have the result 4-15, but we were told to divide 4-5 by 3-7, and consequently we have used a divisor 7 times too large. Hence the quotient is 1-7 of the correct quotient. To correct this, multiply 4-15 by 7. According to preceding principle to multiply a fraction by an integer we must multiply the numerator or divide the denominator. In this instance we will multiply the numerator, and we have as a result 28-15, and, therefore, 4-5 divided by 3-7 equals 28-15.
If, after solving a dozen or more problems in this way, the pupil is required to generalize, the following will be the result: To divide a fraction by a fraction multiply the numerator of the dividend by the denominator of the divisor for a new numerator, and multiply the denominator of the dividend by the numerator of the divisor for a new denominator. Having made this generalization the teacher may take the next step, provided he thinks it is desirable, and lead the pupil to see this result is precisely the same as it would have been had he inverted the terms of his divisor and proceeded as in multiplication. Too much time is devoted to this so-called case. An immense amount of time is wasted, worse than wasted.
We do not wise to be misunderstood in our treatment of dividing a fraction by a fraction, or in our treatment of multiplying a fraction by a fraction. We are aware that other explanations can be offered. We present these explanations because they are based upon the "principles of fractions." We have been careful to estimate the value of these cases. We have been careful to ask the teacher to look to those applications of multiplication and division of fractions that the business man constantly employ. Let the reader bear in mind that nothing is lose in mental discipline, nothing in time; everything works for economy and discipline.
We shall not consume the valuable time of our readers in presenting what are commonly termed complex fractions. In advanced classes there may be found a place for presenting complex fractions which involve addition, subtraction, multiplication and division in long and detailed combinations. Beyond question, there is a disciplinary value that can be attached to the solution of this class of problems. Our experience down not justify us in recommending the rank and file of the teacher profession to recognize these problems as at all essential to the mastery of fractions as required in the majority of our schools. We have purposely omitted those problems in which sign language is a conspicuous factor. Now infrequently in examinations for teachers, this requirement has been emphasized. The result is that what might very properly be called a "dead language" has been called into use, a language that the counting room knows nothing about, needs to know nothing about, doesn't wish to know anything about, would dislike to know anything about.
Notwithstanding the fact that in addition, subtraction, and multiplication of fractions many applications have been made to business transactions, a large number of problems should now be given that involve the use of reason. Reason, in order to procure results, will have to employ addition, subtraction, multiplication, and division. If the pupils have been taught fractions thus far mechanically, they will long for the day to come when they will have finished what is commonly termed "miscellaneous examples." If they have taken delight in using their minds, if they are quick and accurate, they will ask for a larger number of miscellaneous problems than they are given in the ordinary arithmetic. The best problems should be selected from three or four of the very best arithmetics. When a pupil can solve and explain these problems readily, he may rest assured that he has mastered the essential difficulties of arithmetic. It is true that new things will arise, but they will consist largely of new applications to the usages of the business world, and these applications will be made with ease and dispatch provided he masters the language of the new case involving a new business.
Within the past three weeks, from a careful study of two hundred pupils, most of them pupils who have taught school, we find that their difficulties in percentage sustain a close relation to their ignorance of common fractions. In every instance the student who is thoroughly familiar with fractions, who can handle them rapidly and accurately, who can analyze, has little or no difficulty in any of the cases of percentage.
In concluding our work on fractions, we would, therefore, again ask teachers to emphasize the mental exercise. Whenever unsurmountable difficulties present themselves in written work, fall back upon the mental. Insist upon verification in the solution of every problem. We asked for this in one of our articles on mental arithmetic. The business world demands verification, will not recognize the calculator who does not verify. Let the teacher avoid hurry, avoid superficial work in leading young people to master the subject of fractions. This subject has not been over emphasized. Its value has not been exaggerated. Reform in this work is needed.
The writer of these articles invites criticism upon what he has had to say thus far. Let teachers who have had years of experience in teaching fractions express. The writer will be very glad to consider the difficulties encountered by his co-workers.